I know Tywin was in a bit of a tough place when Robb went west but shouldn’t his first priority still have been Kings Landing? If Stannis overthrows Joffrey, Tywin knows Stannis will have Tywin’s head. Sure, staying put places a lot of stress on his men (many might desert) but going west seems like a doomed strategy (but for GRRM’s heavy thumb). Wouldn’t it have made some sense to at least send some reinforcements to KL and/or strong diplomacy with the Tyrells? We don’t really seem that yet.

So I talked about this a lot in my coverage of ACOK. In many ways, Tywin is in a no-win scenario: if he abandons the west, he might arrive in King’s Landing without an army; if he heads west, he might lose the capital, the king, and thus any sense of a political sentiment. My guess is that he was banking on King’s Landing being able to last long enough in a siege situation for him to get to the Westerlands and back.

Splitting his army would have been very dangerous indeed, because in that scenario he wouldn’t have had the numerical advantage against Edmure’s army, which raises the odds that Edmure win a much more decisive victory at the Red Fork and/or go on the offensive and catch Tywin’s army before it can get to King’s Landing. 

Diplomacy with the Tyrells worked out in OTL, but was very much a “heavy thumb” moment. Keep in mind the Tyrells have been the major threat to King’s Landing, blockading grain supplies and besieging it from a distance, and gradually moving up their army to attack directly. Up until Renly unexpectly dies, they have no reason to make a deal with Tywin rather than overthrow his regime.